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Contents:

## Engineering Mathematics

This subject helps a student gain analytical ability in solving mathematics problems by applying them in respective branches of engineering. This is about applying advanced matrix knowledge in engineering problems.

## Download B.Tech 1^{st} Year Maths PDF

Engineering mathematics textbook pdf free download | Download |

first-year engineering mathematics notes | Download |

Engineering mathematics 1 notes free download | Download |

Engineering mathematics 2 pdf | Download |

Engineering mathematics 3 question papers pdf | Download |

Engineering mathematics 1 question papers pdf | Download |

Engineering mathematics 2 Question paper | Download |

## Recommended Books

- Ramana B.V., Higher Engineering Mathematics, TMH, Ist edition
- J.Sinha Roy and S Padhy, A course on ordinary and partial differential Equation, Kalyani Publication, 3rd edition
- Kreyszig E., Advanced Engineering Mathematics, Wiley, 9th edition.
- Shanti Narayan and P.K.Mittal, Differential Calculus, S. Chand, reprint 2009
- Grewal B.S., Higher Engineering Mathematics, Khanna Publishers,36th edition
- Dass H.K., Introduction to engineering Mathematics, S.Chand & Co Ltd, 11th edition
- Ramana B.V., Higher Engineering Mathematics, TMH, 1st edition
- J.Sinha Roy and S Padhy, A course on ordinary and partial differential Equation, Kalyani Publication, 3rd edition
- Chakraborty and Das; Principles of transportation engineering; pHI
- Rangwala SC; Railway Engineering; charotar Publication House, Anand
- Rangwala sc; Bridge Engineering; charotar Publication House, Anand
- Ponnuswamy; Bridge Engineering; TMH
- Kreyszig E., Advanced Engineering Mathematics, Wiley, 9th edition.
- Grewal B.S., Higher Engineering Mathematics, Khanna Publishers, 36th edition
- Dass H.K., Introduction to engineering Mathematics, S.Chand & Co Ltd, 11th edition

#### Syllabus

## Mathematics I:

I: Ordinary Differential Equations :

Basic concepts and definitions of 1st order differential equations; Formation of differential equations; solution of

differential equations: variable separable, homogeneous, equations reducible to homogeneous form, exact differential equation, equations reducible to exact form, linear differential equation, equations reducible to linear form (Bernoulli’s equation); orthogonal trajectories, applications of differential equations.

II: Linear Differential equations of 2nd and higher-order

Second-order linear homogeneous equations with constant coefficients; differential operators; solution of homogeneous equations; Euler-Cauchy equation; linear dependence and independence; Wronskian; Solution of nonhomogeneous equations: general solution, complementary function, particular integral; solution by variation of parameters; undetermined coefficients; higher order linear homogeneous equations; applications.

III: Differential Calculus(Two and Three variables)

Taylor’s Theorem, Maxima, and Minima, Lagrange’s multipliers

IV: Matrices, determinants, linear system of equations

Basic concepts of algebra of matrices; types of matrices; Vector Space, Sub-space, Basis and dimension, linear the system of equations; consistency of linear systems; rank of matrix; Gauss elimination; inverse of a matrix by Gauss Jordan method; linear dependence and independence, linear transformation; inverse transformation ; applications of matrices; determinants; Cramer’s rule.

V: Matrix-Eigen value problems

Eigen values, Eigen vectors, Cayley Hamilton theorem, basis, complex matrices; quadratic form; Hermitian, SkewHermitian forms; similar matrices; diagonalization of matrices; transformation of forms to principal axis (conic section).

## MATHEMATICS-II

I: Laplace Transforms

Laplace Transform, Inverse Laplace Transform, Linearity, transform of derivatives and Integrals, Unit Step function, Dirac delta function, Second Shifting theorem, Differentiation and Integration of Transforms, Convolution, Integral Equation, Application to solve differential and integral equations, Systems of differential equations.

II: Series Solution of Differential Equations

Power series; the radius of convergence, power series method, Frobenius method; Special functions: Gamma function,

Beta function; Legendre’s and Bessel’s equations; Legendre’s function, Bessel’s function, orthogonal functions;

generating functions.

**III: Fourier series, Integrals and Transforms**

Periodic functions, Even and Odd functions, Fourier series, Half Range Expansion, Fourier Integrals, Fourier sine, and cosine transforms, Fourier Transform

**IV: Vector Differential Calculus**

Vector and Scalar functions and fields, Derivatives, Gradient of a scalar field, Directional derivative, Divergence of a vector field, Curl of a vector field.

**V: Vector Integral Calculus**

Line integral, Double Integral, Green’s theorem, Surface Integral, Triple Integral, Divergence Theorem for Gauss, Stoke’s Theorem

**Engineering Mathematics III:**

**UNIT I:** Linear systems of equations:

Rank-Echelon form-Normal form – Solution of linear systems – Gauss elimination – Gauss Jordon- Gauss Jacobi and Gauss Seidel methods. Applications: Finding the current in electrical circuits.

**UNIT II: **Eigenvalues – Eigenvectors and Quadratic forms:

Eigen values – Eigen vectors– Properties – Cayley-Hamilton theorem Inverse and powers of a matrix by using Cayley-Hamilton theorem- Diagonalization- Quadratic forms- Reduction of quadratic form to canonical form – Rank – Positive, negative and semi definite – Index – Signature. Applications: Free vibration of a two-mass system.

**UNIT III:** Multiple integrals:

Curve tracing: Cartesian, Polar and Parametric forms. Multiple integrals: Double and triple integrals – Change of variables –Change of order of integration. Applications: Finding Areas and Volumes.

**UNIT IV:** Special functions:

Beta and Gamma functions- Properties – Relation between Beta and Gamma functions- Evaluation of improper integrals.

Applications: Evaluation of integrals.

**UNIT V:** Vector Differentiation:

Gradient- Divergence- Curl – Laplacian and second-order operators -Vector identities. Applications: Equation of continuity, potential surfaces

**UNIT VI:** Vector Integration:

Line integral – Work is done – Potential function – Area- Surface and volume integrals Vector integral theorems: Greens, Stokes and Gauss Divergence theorems (without proof) and related problems.

Applications: Work is done, Force.

## Important Questions

- Give examples of Hermitian, skew-Hermitian and unitary matrices that have entries with non-zero imaginary parts.
- Restate the results on transpose in terms of the conjugate transpose.
- Show that for any square matrix A, S = A+A* 2 is Hermitian, T = A−A ∗ 2 is skew-Hermitian, and A = S + T.
- Show that if A is a complex triangular matrix and AA∗ = A∗A then A is a diagonal matrix